DSpace at VNU: Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels

Composite Structures 94 (2012) 2465–2473 Contents lists available at SciVerse ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels Dao Huy Bich a, Dao Van Dung a, Vu Hoai Nam b,⇑ a b Vietnam National University, Hanoi, Viet Nam Faculty of Civil Engineering, University of Transport Technology, Ha Noi, Viet Nam a r t i c l e i n f o Article history: Available online 28 March 2012 Keywords: Functionally graded material (FGM) Dynamical analysis Critical dynamic buckling load Vibration Cylindrical panel Stiffeners a b s t r a c t Based on the classical shell theory with the geometrical nonlinearity in von Karman–Donnell sense and the smeared stiffeners technique, the governing equations of motion of eccentrically stiffened functionally graded cylindrical panels with geometrically imperfections are derived in this paper The characteristics of free vibration and nonlinear responses are investigated The nonlinear dynamic buckling of cylindrical panel acted on by axial loading is considered The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth Some numerical results are given and compared with the ones of other authors Ó 2012 Elsevier Ltd All rights reserved Introduction Functionally graded materials (FGMs) are composite materials which have mechanical properties varying continuously from one surface to the other of structure The concept of functionally graded material was proposed in 1984 [1] and it is often used in heat-resistance structure as elements in aerospace and nuclear reactors [2] Today, the application of this material is getting varied, so the problems of static and dynamic behaviors of structures such as FGM plates and shells have been properly noticed For dynamical analysis of FGM shells, many studies have been focused on the characters of vibration and behavior of buckling of shells Ng et al [3] and Darabi et al [4] presented respectively linear and nonlinear parametric resonance analyses for FGM cylindrical shells Loy et al [5] and Pradhan et al [6] studied the free vibration characteristics of FGM cylindrical shells By using Galerkin technique together with Ritz type variational method, Sofiyev [7] and Sofiyev and Schnack [8] obtained critical parameters for cylindrical thin shells under linearly increasing dynamic torsional loading, and under a periodic axial impulsive loading By using a higher order shell theory and a finite element solving method, Shariyat [9] investigated nonlinear dynamic buckling problems of axially and laterally preloaded FGM cylindrical shells under transient thermal shocks Geometrical imperfection effects were also included in his research Using the similar method, he also presented a dynamic buckling analysis for FGM cylindrical ⇑ Corresponding author E-mail address: hoainam.vu@utt.edu.vn (V.H Nam) 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd All rights reserved http://dx.doi.org/10.1016/j.compstruct.2012.03.012 shells under complex combinations of thermo–electro-mechanical loads [10] Huang and Han [11] presented nonlinear dynamic buckling problems of functionally graded cylindrical shells subjected to time-dependent axial load by using Budiansky–Roth dynamic buckling criterion [12] Various effects of the inhomogeneous parameter, loading speed, dimension parameters; environmental temperature rise and initial geometrical imperfection on nonlinear dynamic buckling were discussed Liew et al [13] presented the nonlinear vibration analysis for layered cylindrical panels containing FGMs and subjected to a temperature gradient arising from steady heat conduction through the panel thickness Ganapathi [14] studied the dynamic stability behavior of a clamped FGMs spherical shell structural element subjected to external pressure load He solved the governing equations employing the Newmark’s integration technique coupled with a modified Newton–Raphson iteration scheme Sofiyev [15–17] studied the vibration and buckling of the FGM truncated conical shells under dynamic axial loading Based on first-order shear deformation theory, the dynamic thermal buckling behavior of functionally graded spherical caps is studied by Prakash et al [18] Dynamic buckling of functionally graded materials truncated conical shells subjected to normal impact loads is discussed by Zang and Li [19] For FGM shallow shells, Alijani et al [20], Chorfi and Houmat [21] and Matsunaga [22] investigated nonlinear forced vibrations of FGM doubly curved shallow shells with a rectangular base Nonlinear dynamical analysis of imperfect functionally graded material shallow shells subjected to axial compressive load and transverse load was studied by Bich and Long [23] and Dung and Nam [24] The motion, stability and compatibility equations of these 2466 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 structures were derived using the classical shell theory The nonlinear transient responses of cylindrical and doubly-cuvred shallow shells subjected to excited external forces were obtained and the dynamic critical buckling loads were evaluated based on the displacement responses using Budiansky–Roth dynamic buckling criterion [12] However, there are very little researches on nonlinear dynamic problems of imperfect eccentrically stiffened functionally graded shells Recently, Najafizadeh et al [25] studied statical buckling behaviors of FGM cylindrical shell Bich et al [26] have studied the nonlinear statical postbuckling of eccentrically stiffened functionally graded plates and shallow shells Following the idea of Ref [26], this paper establishes dynamics governing equations and investigates nonlinear vibration and dynamic buckling of imperfect reinforced FGM cylindrical panel It shows the influences of stiffener, of volume-fraction index, of initial imperfection and of geometrical parameters to the dynamic characteristics of panels According to the classical shell theory and geometrical nonlinearity in von Karman–Donnell sense, the strains at the middle surface and curvatures are related to the displacement components u, v, w in the x1, x2, z coordinate directions as [28]  2 @u @w @2w ỵ ; v1 ¼ ; @x1 @x1 @x1  2 @v 1 @w @2w ẳ wỵ ; v2 ¼ ; @x2 @x2 R @x2 e01 ¼ e02 c012 ẳ @u @ v @w @w ỵ ỵ ; @x2 @x1 @x1 @x2 where R is radius of the cylindrical shell The strains across the shell thickness at a distance z from the mid-surface are e1 ¼ e01 À zv1 ; e2 ¼ e02 À zv2 ; c12 ¼ c012 À 2zv12 ; @ e01 @ e02 @ c012 ỵ ẳ @x2 @x1 @x1 @x2 2.1 Functionally graded material Functionally graded material in this paper, is assumed to be made from a mixture of ceramic and metal with the volume-fractions given by a power law V m ỵ V c ẳ 1; where h is the thickness of panel; k P is the volume-fraction index; z is the thickness coordinate and varies from Àh/2 to h/2; the subscripts m and c refer to the metal and ceramic constituents respectively According to the mentioned law, the Young modulus and the mass density can be expressed in the form Ezị ẳ Em V m ỵ Ec V c ẳ Em ỵ Ec Em Þ ð1Þ the Poissons’s ratio m is assumed to be constant 2.2 Constitutive relations and governing equations Consider a functionally graded cylindrical thin panel in-plane edges a and b The panel is reinforced by eccentrically longitudinal and transversal stiffeners The cylindrical panel is assumed to have a relative small rise as compared with its span Let the (x1, x2) plane of the Cartesian coordinates overlaps the rectangular plane area of the panel Note that the middle surface of the panel generally is defined in terms of curvilinear coordinates, but for the cylindrical panel, so the Cartesian coordinates can replace the curvilinear coordinates on the middle surface (see Fig 1) s1 (i) Longitudinal Stiffeners z1 À @2w @2w @2w À : @x21 @x22 R @x21 ð4Þ ð5aÞ and for stiffeners st st r ¼ E0 e1 ; r ẳ E0 e2 5bị where E0 is Youngs modulus of ring and stringer stiffeners Taking into account the contribution of stiffeners by the smeared stiffeners technique and omitting the twist of stiffeners and integrating the stress–strain equations and their moments through the thickness of the panel, we obtain the expressions for force and moment resultants of an ES-FGM cylindrical panel   E0 A1 N1 ẳ A11 ỵ e1 ỵ A12 e02 B11 ỵ C ịv1 À B12 v2 ; s1   E A2 e2 B12 v1 B22 ỵ C ịv2 ; N2 ẳ A12 e01 ỵ A22 ỵ s2 6ị N12 ẳ A66 c012 2B66 v12 ;   E0 I1 v1 À D12 v2 ; M ẳ B11 ỵ C ịe01 ỵ B12 e02 D11 ỵ s1   E0 I v2 ; M ẳ B12 e01 ỵ B22 ỵ C ịe02 D12 v1 D22 ỵ s2 M 12 ¼ B66 c012 À 2D66 v12 ; a x2 !2 Ezị e1 ỵ me2 ị; m2 Ezị ẳ e2 ỵ me1 ị; m2 Ezị ẳ c ; 21 ỵ mị 12 b d1 @2w @x1 @x2 rsh ẳ ssh 12 z 3ị Hooks stressstrain relation is applied for the shell rsh  k 2z ỵ h ; V c ẳ V c zị ¼ 2h h1 @2w ; @x1 @x2 v12 ¼ From Eq (3) the strains must be relative in the deformation compatibility equation Eccentrically stiffened FGM cylindrical panels (ES-FGM cylindrical panels)  k 2z ỵ h ; 2h  k 2z ỵ h qzị ẳ qm V m ỵ qc V c ẳ qm ỵ qc qm ị ; 2h ð2Þ x1 h2 z d2 s2 (ii) Transversal Stiffeners Fig Configuration of an eccentrically stiffened cylindrical panel z2 ð7Þ 2467 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 where Aij, Bij, Dij (i, j = 1, 2, 6) are extensional, coupling and bending stiffnesses of the panel without stiffeners Z h=2 Z h=2 EðzÞ E1 Ezịm E1 m A11 ẳ A22 ẳ dz ẳ ; A12 ¼ dz ¼ ; 2 À m2 À m2 Àh=2 À m Àh=2 À m Z h=2 Ezị E1 A66 ẳ dz ẳ ; 21 ỵ mị h=2 21 ỵ mị Z h=2 Z h=2 zEzị E2 zEzịm E2 m B11 ẳ B22 ẳ dz ¼ ; B ¼ dz ¼ ; 12 2 À m À m À m À m2 Àh=2 Àh=2 Z h=2 zEðzÞ E2 dz ẳ ; B66 ẳ 21 ỵ m ị 21 þ mÞ Àh=2 Z h=2 Z h=2 z Ezị E3 z Ezịm E3 m D11 ẳ D22 ẳ dz ¼ ; D ¼ dz ¼ ; 12 2 À m À m À m À m2 Àh=2 Àh=2 Z h=2 z Ezị E3 D66 ẳ dz ẳ ; 21 ỵ m ị 21 ỵ mị h=2 8ị with   Ec Em Ec Em ịkh E1 ẳ Em ỵ h; E2 ẳ ; kỵ1  2k ỵ 1ịk þ 2Þ ! Em 1 h ; E3 ẳ ỵ ỵ Ec Em ị k þ k þ 4k þ 12 3 d1 h1 d2 h2 I1 ẳ ỵ A1 z21 ; I2 ẳ ỵ A2 z22 : 12 12 Substituting Eq (10) into Eq (7) yields M ¼ BÃ11 N1 ỵ B21 N2 D11 v1 D12 v2 ; M ẳ B12 N1 ỵ B22 N2 DÃ21 v1 À DÃ22 v2 ; M 12 ¼ BÃ66 N12 À 2DÃ66 v12 ; where E0 I1 À ðB11 þ C ÞBÃ11 À B12 BÃ21 ; s1 E0 I2 ẳ D22 ỵ B12 B12 B22 ỵ C ịB22 ; s2 ẳ D12 B11 ỵ C ịB12 B12 B22 ; D11 ẳ D11 þ DÃ22 DÃ12 DÃ66 ¼ D66 À B66 BÃ66 : The nonlinear equations of motion of a cylindrical thin panel based on the classical shell theory and the assumption (Refs 2 [4,8,27]) u ( w and v ( w, q1 @@t2u ! 0, q1 @@t2v ! are given by @N1 @N12 ỵ ẳ 0; @x1 @x2 9ị @N12 @N2 ỵ ẳ 0; @x1 @x2 @ M1 @ M12 @ M2 @2w @2w ỵ2 ỵ ỵ N ỵ 2N12 @x1 @x2 @x1 @x2 @x21 @x22 @x1 ỵ N2 E A1 z E0 A2 z2 C1 ¼ ; C2 ¼ ; s1 s2 h1 ỵ h h2 ỵ h ; z2 ẳ : z1 ẳ 2 10ị q1 ẳ ỵ q0 B21 ẳ A11 B12 A12 B11 ỵ C ị; B66 B66 ẳ : A66 14ị h=2 qzịdz ỵ q0   A1 A2 ; ỵ s1 s2     A1 A2 q qm ẳ qm ỵ c ỵ h s1 s2 kỵ1 with q0 = qm for metal stiffener, q0 = qc for ceramic stiffener The first two of Eq (14) are satisfied automatically by choosing a stress function u as N1 ¼ @2u ; @x22 N2 ¼ @2u ; @x21 N12 ¼ À @2u : @x1 @x2 @4/ @4u @4u @4w ỵ A66 2A12 ỵ A22 ỵ B21 4 2 @x1 @x2 @x1 @x1 @x2 À Á @4w @4w @2w ỵ B11 ỵ B22 2B66 ỵ B12 ỵ 2 @x1 @x2 @x2 R @x21 !2 @2w @2w @2w ¼ À ; @x1 @x2 @x1 @x22 ð11Þ ð15Þ The substitution of Eq (10) into the compatibility Eqs (4) and (12) into the third of Eq (14), taking into account expressions (2) and (15), yields a system of equations Ẫ11 ¼ BÃ22 ¼ Ẫ11 ðB22 ỵ C ị A12 B12 ; B12 ẳ A22 B12 A12 B22 ỵ C ị; Z h=2 A11 where B11 ẳ A22 B11 ỵ C ị A12 B12 ; @2w @2w ỵ N2 þ q0 ¼ q1 ; @x2 R @t where In above relations (6), (7) and (9) the quantityE0 is the Young modulus in the axial direction of the corresponding stiffener which is assumed identical for both types of stiffeners, it takes the value E0 = Em if the full metal stiffeners are put at the metal-rich side of the panel and conversely E0 = Ec if the full ceramic ones at the ceramic-rich side Such FGM stiffened cylindrical panels provide continuity within panel and stiffeners and can be easier manufactured The spacings of the longitudinal and transversal stiffeners are denoted by s1 and s2 respectively The quantities A1, A2 are the cross-section areas of stiffeners and I1, I2, z1, z2 are the second moments of cross section areas and the eccentricities of stiffeners with respect to the middle surface of panel respectively The strain-force resultant relations reversely are obtained from Eq (6)     E A1 E0 A2 ; Ẫ22 ¼ ; A11 þ A22 þ D s1 D s2 A12 AÃ12 ¼ ; AÃ66 ¼ ; A66 D    E0 A1 E A2 A22 ỵ A212 ; D ẳ A11 ỵ s1 s2 13ị D21 ẳ D12 B12 B11 B22 ỵ C ịB21 ; and e01 ẳ A22 N1 A12 N2 ỵ B11 v1 ỵ B12 v2 ; e02 ẳ A11 N2 A12 N1 ỵ B21 v1 ỵ B22 v2 ; c012 ẳ A66 ỵ 2B66 v12 ; 12ị 16ị @4w @2w @4w @4w ỵ D11 ỵ D12 ỵ D21 ỵ 4D66 ỵ D22 2 @x1 @x1 @x2 @x2 @t 4 À Á @ u @ u @ u @2u à À BÃ21 B11 ỵ B22 2B66 B 12 @x1 @x21 @x22 @x42 R @x21 q1 À @2u @2w @2u @2w @2u @2w ỵ2 ẳ q0 ; 2 @x1 @x2 @x1 @x2 @x1 @x22 @x2 @x1 ð17Þ 2468 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 For initial imperfection ES-FGM panels: The initial imperfection of the panel considered here can be seen as a small deviation of the panel middle surface from the perfect shape, also seen as an initial deflection which is very small compared with the panel dimensions, but may be compared with the panel wall thickness Let w0 = w0(x1, x2) denote a known small imperfection, proceeding from the motion Eqs (16) and (17) of a perfect FGM cylindrical panel and following to the Volmir’s approach [27] for an imperfection panel we can formulate the system of motion equations for an imperfect eccentrically stiffened functionally graded cylindrical panel (imperfect ES-FGM cylindrical panel) as AÃ11 Á @4u @4u À à @4u @ w w0 ị ỵ A66 2A12 þ AÃ22 þ BÃ21 2 @x41 @x1 @x1 @x2 @x2 @ ðw À w0 Þ ¼ 0; R @x21 À r0 h 2mpx1 2npx2 mpx1 npx2 ỵ u2 cos u3 sin sin a b a b x22 ; 23ị where denote u1 ẳ n2 k2 f ; 32m2 AÃ11 m2 f u2 ¼ ; 32n2 k2 Ẫ22 h i À Á B21 m4 ỵ B11 ỵ B22 2B66 m2 n2 k2 ỵ B12 n4 k4 pa 1R m2 u3 ẳ : A11 m4 ỵ A66 2A12 m2 n2 k2 ỵ A22 n4 k4 24ị Á @ ðw À w0 Þ @ ðw w0 ị ỵ B11 ỵ B22 2BÃ66 @x42 @x21 @x22 3 !2 !2 2 2 2 @ w @ w @ w @ w @ w @ w 0 5ỵ4 @x1 @x2 @x1 @x2 @x21 @x22 @x1 @x22 ỵ B12 ỵ u ẳ u1 cos Substituting the expressions (21)–(23) into Eq (19) and applying Galerkin method to the resulting equation yield ! À Á B2 8mnk2 B d1 d2 ðf À f0 Þf þ H f À f02 ðf À f0 Þ þ M€f þ D þ A 3p2 A ð18Þ À a2 h 4q a4 ỵ K f f02 f À r0 m2 f À d1 d2 ẳ 0; p mnp 25ị where denote @ ðw À w0 Þ @2w @ ðw w0 ị q1 ỵ D11 ỵ D12 ỵ D21 ỵ 4D66 @x1 @x21 @x22 @t þ DÃ22 2 2 2 a4 p4 q1 ; A ẳ A11 m4 ỵ A66 2A12 m2 n2 k2 ỵ A22 n4 k4 ; Á @4u @ ðw À w0 Þ @4u À B21 B11 ỵ B22 2B66 @x1 @x21 @x22 @x42 À BÃ12 M¼ @ u 1@ u @ u@ w @ u @ w @ u@ w ỵ2 ẳ q0 ; 19ị @x1 @x2 @x1 @x2 @x21 @x22 @x42 R @x21 @x22 @x21 where w is a total deflection of panel Hereafter, the couple of Eqs (16) and (17) or of Eqs (18) and (19) are used to investigate the nonlinear vibration and dynamic stability of panels They are nonlinear equations in terms of two dependent unknowns w and u À Á a2 B ẳ B21 m4 ỵ B11 ỵ B22 2B66 m2 n2 k2 ỵ B12 n4 k4 m2 ; p R À Á D ¼ DÃ11 m4 ỵ D12 ỵ D21 ỵ 4D66 m2 n2 k2 þ DÃ22 n4 k4 ; " #   2mnk2 BÃ21 BÃ12 a2 n2 k2 d1 d2 ; À þ H¼ 3p2 Ẫ11 Ẫ22 6p4 mn Ẫ11 R ! m4 n4 k4 a ; kẳ ; ỵ Kẳ 16 A22 b A11 26ị d1 ẳ 1ịm 1; Nonlinear dynamic analysis d2 ẳ 1ịn 1: 3.1 Solution of the problem The obtained Eq (25) is the governing equation for dynamic analysis of ES-FGM cylindrical panels in general Based on this equation the non-linear vibration of perfect and imperfect FGM cylindrical panels can be investigated and the dynamic buckling analysis of panels under various loading cases can be performed Particularly for a plate, R = is taken in Eqs (25) and (26) Suppose that an imperfect ES-FGM cylindrical panel is simply supported and subjected to uniformly distributed pressure of intensity q0 and in plane compressive load of intensities r0 at its cross-section (in Pa) Thus the boundary conditions considered in the current study are w ¼ 0; M ¼ 0; N1 ¼ Àr h; w ¼ 0; M ¼ 0; N2 ¼ 0; N 12 ¼ 0; N 12 ¼ 0; at at x1 ¼ 0; x2 ¼ 0; b: a; 3.2 Vibration analysis ð20Þ where a and b are the lengths of in-plane edges of the panel The mentioned conditions (20) can be satisfied identically if the buckling mode shape is represented by w ẳ f tị sin mpx1 npx2 sin ; a b ð21Þ where f(t) is time dependent total amplitude and m, n are numbers of haft wave in axial and circumferential directions, respectively The initial-imperfection w0 is assumed to have similar form of the panel deflection w, i.e w0 ¼ f0 sin mpx1 npx2 sin ; a b ð22Þ where f0 is the known initial amplitude Substituting Eqs (21) and (22) into Eq (18) and solving obtained equation for unknown u lead to Consider an imperfect ES-FGM cylindrical panel acted on by an uniformly distributed excited transverse load q0 = Q sin Xt and r0 = 0, the non-linear Eq (25) has of the form ! À Á B2 8mnk2 B f f0 ị ỵ d1 d2 f f0 ịf ỵ H f f02 Mf ỵ D ỵ A 3p2 A 4a4 ỵ K f f02 f ẳ d d Q sin Xt: p mn ð27Þ By using Eq (27), three aspects are taken into consideration: fundamental frequencies of natural vibration of ES-FGM panel and FGM panel without stiffeners, frequency–amplitude relation of non-linear free vibration and non-linear response of ES-FGM panel The non-linear dynamical responses of ES-FGM panels can be obtained by solving this equation combined with initial conditions to be assumed as f 0ị ẳ 0; f_ 0ị ẳ by using the RungeKutta iteration schema 2469 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 If the vibration is free and linear, Eq (27) leads to m2 a2 h ! p2 B Mf ỵ D ỵ f f0 ị ẳ 0; A ð28Þ from which the fundamental frequencies of natural vibration of imperfect ES-FGM cylindrical panels can be determined by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u1 B2 t : Dỵ xL ẳ M A 29ị rupper ẳ where ! B2 ; Dỵ M A xNL ẳ xL ỵ 8H2 3H g þ 32 g2 3px2L 4xL ð35Þ p2 Dþ m2 a2 h ! B2 ; A ð36Þ p2 ð37Þ Suppose axial load varying linearly on time r0 = ct(c (in Pa/s) is a loading speed) and introduce parameters: H3 ¼ K : M ð31Þ Seeking solution as f(t) = gcosxt and applying procedure like Galerkin method to Eq (30), the frequency–amplitude relation of non-linear free vibration is obtained  8mnk2 B d1 d2 ỵ H f ỵ Kf : 3p2 A b ẳ 8mnk B d1 d2 ỵ H: H 3p A where denoting ! 8mnk2 B d ỵ H ; d M 3p A ỵ ! ^2 B2 H ẳ 2 Dỵ ; m a h A 4K f ỵ H f ỵ H f ỵ H f ẳ 0; H2 ¼ B2 A and the lower static buckling load is found using the condition dr0 ¼ 0, it follows df rlower H1 ¼ x2L ¼ Dỵ ! From Eq (35), the upper static buckling load can be determined by putting f = The equation of non-linear free vibration of a perfect panel can be obtained from (27) 30ị r0 ẳ ! 12 ; 32ị where xNL is the non-linear vibration frequency and g is the amplitude of non-linear vibration D¼ D h f n¼ ; h ; B ; h f0 n0 ¼ ; h B¼ A ¼ Ah; s¼ H¼ H h ; K¼ K ; h r0 ct ¼ ; r scr r scr ð38Þ where rscr = minrupper vs (m, n) The non-dimension form of Eq (33) is written as d n ỵ S1 ds2 " Dỵ i ỵk n2 n20 n 3.3 Nonlinear dynamic buckling analysis B2 ! A p2 ðn À n0 ị ỵ k4 hb rscr 8mnk2 B d1 d1 n n0 ịnỵỵH n2 n20 3p2 A m2 k ns ẳ 0: 39ị where Investigate the non-linear dynamic buckling of imperfect ESFGM cylindrical panels in some cases of active loads varying as linear function of time The aim of considered problems is to seek the critical dynamic buckling loads They can be evaluated based in the displacement responses obtained from the motion Eqs (25) and (26) The criterion suggested by Budiansky and Roth [12] is employed here as it is widely accepted This criterion is based on that, for large value of loading speed, the amplitude-time curve of obtained displacement response increases sharply depending on time and this curve obtains a maximum by passing from the slope point and at the corresponding time t = tcr the stability loss occurs Here t = tcr is called critical time and the load corresponding to this critical time is called dynamic critical buckling load Consider an ES-FGM cylindrical panel subjected to axial load r0(t) In this case q0 = 0, Eq (25) gives Mf ỵ D þ ! À Á B2 8mnk2 B ðf À f0 ị ỵ d1 d2 f f0 ịf ỵ H f À f02 A 3p A À Á m2 a2 h r0 tị f ẳ ỵ K f À f02 f À p ð33Þ Omitting the term of inertia and putting f0 = in Eq (33), yields an equation for determining the static critical load of ES-FGM cylindrical panels as m2 a2 h p2 r0 f ẳ ! ! B2 8mnk2 B fỵ d1 d2 ỵ H f ỵ Kf : Dỵ A 3p A ð34Þ Taking f – 0, i.e considering the panel after the lost of stability we obtain S1 ¼ p2 r3scr h : b c2 q1 ð40Þ Solving Eq (39) by Runge–Kutta method and applying Budiansky–Roth criterion, the critical value sdcr, the dynamic critical time tdcr ¼ sdcrcrscr and dynamical buckling load rdcr = ctdcr respectively are obtained Numerical results and discussions 4.1 Validation of the present formulation In this section, first of all, the qffiffiffifficomparison on the fundamental ~ ¼ xL h qE c (xL is calculated from Eq (29)) frequency parameter x c given by the present analysis with the results of Alijani et al.[20] based on the Donnell’s nonlinear shallow-shell theory, Chorfi and Houmat [21] based on the first-order shear deformation theory and Matsunaga [22] based on the two-dimensional (2D) higher-order theory for the perfect unreinforced FGM cylindrical panel Àa Á ¼ 1; ¼ 0:1 with simply supported movable edges is suggested b The material properties in Refs [20–22] are aluminium and alumina, i.e Em = 70.109 N/m2, qm = 2702 kg/m3 and Ec = 380.109 N/m2, qc = 3800 kg/m3 respectively The Poisson’s ratio is chosen to be 0.3 As can be observed in Table 1, a very good agreement is obtained in this comparison study Next, the present frequency xL (in Eq (29)) is compared with the result of Szilard [29] and Troitsky [30] based on the classical assumptions of small deformations and thin plates Consider a simply supported homogeneous plate that is biaxial stiffened with multiple stiffeners (see Fig 2) As shown in Table 2, a good agreement can be witnessed 2470 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 Table ~ with results reported by Alijani et al [20], Chorfi and Houmat [21] Comparison of x and Matsunaga [22] a/R k Present Ref [20] Ref [21] Ref [22] 0.0597 0.0506 0.0456 0.0396 0.0381 0.0597 0.0506 0.0456 0.0396 0.0380 0.0577 0.0490 0.0442 0.0383 0.0366 0.0588 0.0492 0.0430 0.0381 0.0364 FGM cylindrical panel 0.5 0.0648 0.5 0.0553 0.0501 0.0430 10 0.0409 0.0648 0.0553 0.0501 0.0430 0.0408 0.0629 0.0540 0.0490 0.0419 0.0395 0.0622 0.0535 0.0485 0.0413 0.0390 FGM plate 0 0.5 10 Table The fundamental frequencies of natural vibration (rad/s) of FGM cylindrical panels R (m) k Unreinforced (m, n) Reinforced (m, n) 0.2 10 1172.51 982.14 822.19 783.56 (1, (1, (1, (1, 3) 3) 3) 3) 1571.27 1435.02 1266.54 1224.47 (1, (1, (1, (1, 2) 2) 2) 2) 0.2 10 803.92 686.91 556.39 519.90 (1, (1, (1, (1, 2) 2) 2) 2) 1192.51 1128.40 1011.97 924.63 (1, (1, (1, (1, 2) 2) 1) 1) 0.2 10 622.96 524.39 435.45 413.06 (1, (1, (1, (1, 2) 2) 2) 2) 930.82 812.67 647.97 599.93 (1, (1, (1, (1, 1) 1) 1) 1) 0.2 10 515.55 438.11 353.51 325.48 (1, (1, (1, (1, 1) 2) 1) 1) 551.26 494.97 427.52 411.30 (1, (1, (1, (1, 1) 1) 1) 1) 0.2 10 197.11 162.11 139.79 135.38 (1, (1, (1, (1, 1) 1) 1) 1) 376.11 364.17 361.77 364.92 (1, (1, (1, (1, 1) 1) 1) 1) 1.5 10 0.6m 0.41 m 0.0127 m 0.02222 m (plates) E=211GPa =0.3 =7830 kg/m3 0.00633 m 0.02222 m 0.0127 m Fig Configuration of an eccentrically stiffened plate Table Comparison of present frequency (Hz) with results reported by Szilard [29] and Troitsky [30] Mode Present Refs [29,30] 10 15 20 357.81 708.34 1248.97 1417.82 1440.38 2888.10 4927.78 5679.99 381.9 719.9 1339.8 1356.1 1512.4 3066.7 5029.2 5589.8 4.2 Vibration results To illustrate the proposed approach to eccentrically stiffened FGM cylindrical panels, the panels considered here are cylindrical panels and plates with in-plane edges a = b = 1.5 m; h = 0.008 m; f0 = The panels are simply supported at all its edges The combination of materials consists of aluminum Em = 70  109 N/m2; qm = 2702 kg/m3 and alumina Ec = 380  109 N/m2, qc = 3800 kg/ m3 The Poisson’s ratio is chosen to be 0.3 for simplicity Material of reinforced stiffeners has elastic modulus E = 380.109 N/m2; q = 3800 kg/m3 The height of stiffeners is equal to 30 mm, its width mm, the spacing of stiffeners s1 = s2 = 0.15 m, the eccentricities of stiffeners with respect to the middle surface of panel z1 = z2 = 0.019 m 4.2.1 Results of fundamental frequencies of natural vibration The obtained results in Table show that the effect of stiffeners on fundamental frequencies of natural vibration xL (xL is calculated from Eq (29)) is considerable Obviously the natural frequencies of unreinforced and reinforced FGM cylindrical panels observed to be dependent on the constituent volume fractions, they decrease when increasing the power index k, furthermore with greater value k the effect of stiffeners is observed to be stronger This is completely reasonable because the lower value is the elasticity modulus of the metal constituent 4.2.2 Results of frequency–amplitude of non-linear free vibration Fig shows the relation frequency–amplitude of non-linear free vibration of reinforced and unreinforced panel (calculated from Eq (32)) with m = 1, n = As expected the non-linear vibration frequencies of reinforced panels are greater than ones of unreinforced panels 4.2.3 Non-linear response results For obtaining the non-linear dynamical responses of FGM cylindrical panel acted on by the harmonic uniformly load q0(t) = Qsin(Xt) with Q =  103 N/m2, X = 975 rad/s and X = 950 rad/s, the Eq (27) is solved using Runge–Kutta method Fig shows non-linear responses of ES-FGM cylindrical panel In this case, exited frequencies are near to fundamental frequencies of natural vibration x = 1011.97 rad/s (see Table 3) From obtained results, the interesting phenomenon is observed like the harmonic beat phenomenon of a linear vibration, in which the amplitude of beats of reinforced panels increased rapidly when the exited frequency approached the natural frequency When the exited frequencies X = 500 rad/s and X = 600 rad/s are away from the natural frequencies of ES-FGM cylindrical panel The obtained non-linear dynamical responses are shown in Fig It shows that, the harmonic beat phenomenon does not appear as in the previous case The amplitude of beats of reinforced panels increased slowly when the exited frequency is close to the natural frequency Fig shows the Influence of initial imperfection with amplitudes f0 = 0, f0 = 10À5 and f0 =  10 À5 m on the non-linear responses of ES-FGM cylindrical panel The initial imperfection f0 has a slight influence to the nonlinear response of panel 2471 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 1.5E+3 ω 8.0E-4 (rad/s) NL R=3m, k=5, q 0=5000sin(500t) f(m) R=10m, k=0.2 6.0E-4 1.2E+3 4.0E-4 Reinforced 9.0E+2 Unreinforced 2.0E-4 0.0E+0 6.0E+2 -2.0E-4 R=10m, k=5 3.0E+2 -4.0E-4 0.0E+0 0.0E+0 η (m) 3.0E-2 6.0E-2 9.0E-2 1.2E-1 -6.0E-4 1.5E-1 0.6 f(m) Ω=975(rad/s) f0 =1e-5 f =5e-5 0.015 0.03 0.045 t(s) 0.06 Fig Influence of initial imperfection on non-linear responses Fig Frequency–amplitude relation 1.2E-2 Perfect Ω=950(rad/s) ξ Unreinforced Panel 0.5 8.0E-3 0.4 4.0E-3 R=3m, k=1 m=5, n=2 0.3 0.0E+0 m=6, n=1 0.2 m=4, n=3 -4.0E-3 m=5, n=1 0.1 -8.0E-3 R=3m, k=5, q 0=5000sin(Ωt) -1.2E-2 0.2 t (s) f(m) Ω=500 (rad/s) τ 9.50E-01 1.00E+00 1.05E+00 1.10E+00 0.4 Fig Effect of buckling mode shapes on load–deflection curve of unreinforced panel Fig Nonlinear response of ES-FGM cylindrical panel 9.0E-4 9.00E-01 1.2 Ω=600 (rad/s) 6.0E-4 3.0E-4 0.8 0.0E+0 0.6 -3.0E-4 0.4 -6.0E-4 0.2 ξ Reinforced Panel R=3m, k=1 m=2, n=2 m=3, n=2 R=3m, k=5, q 0=5000sin(Ωt) -9.0E-4 0.025 0.05 0.075 t (s) 0.1 Fig Nonlinear response of FGM cylindrical panel 4.3 Nonlinear dynamic buckling results To evaluate the effectiveness of the reinforcement of stiffener in the nonlinear dynamic buckling problem, we consider the case of imperfect ES-FGM cylindrical panel subjected to an axial compressive load The critical dynamic buckling loads is determined by solving Eq (39) and applying Budiansky–Roth criterion Materials and structures used in this section are the same in the previous section Figs and show the effect of buckling mode shapes on load – deflection curve of reinforced and unreinforced FGM cylindrical panel subjected to an axial compressive load with the power law index k = 1, R = m and compressive load r0 = 1.5  109 t Clearly, the smallest critical dynamic buckling load corresponds to the buckling mode shape m = 5, n = in the case of unreinforced panel and m = 2, n = in the case of reinforced panel This figure also shows that there is no definite point of instability as in static analysis Rather, 9.00E-01 m=1, n=2 m=2, n=1 τ 1.10E+00 1.30E+00 1.50E+00 1.70E+00 Fig Effect of buckling mode shapes on load–deflection curve of reinforced panel there is a region of instability where the slope of n vs s curve increases rapidly In this paper, the critical parameter scr is taken   ¼ as an intermediate value satisfying the condition dds2n s¼scr Table shows the critical loads of two cases of reinforcement and unreinforcement cylindrical panel The results show that the reinforcement by stiffeners has large effect in the dynamic stability problems of cylindrical panels under axial compressive load With the same input parameters, effectiveness of reinforcement increases as the curvature radius or the power index increases Table also considers the effect of loading speed to the dynamic buckling load; the results show that the dynamic buckling loads increases when the loading speed increases Fig shows the influence of initial imperfection amplitude f0 on the non-linear buckling of ES–FGM Cylindrical panel Clearly, the initial imperfection strongly influences on the critical dynamic buckling loads of ES-FGM cylindrical panel subjected to an axial compressive load 2472 D.H Bich et al / Composite Structures 94 (2012) 2465–2473 Table Nonlinear critical buckling loads of the cylindrical panels subjected to an axial compressive load (Â108 N/m2) R (m) k Unreinforced Static (m, n) Reinforced Dynamic (m, n) Static (m, n) c = 1.5  109 c =  109 Dynamic (m, n) c = 1.5  109 c =  109 0.2 10 5.1667 3.3323 1.9971 1.7111 (5, (5, (5, (5, 2) 2) 2) 1) 5.1945 (5, 3.3795 (5, 2.0700 (5, 1.7895 (5, 2) 2) 2) 1) 5.2125 3.4016 2.0960 1.8160 (5, (5, (5, (5, 2) 2) 2) 1) 9.5082 7.1505 5.0807 4.6866 (2, (2, (2, (2, 2) 2) 2) 2) 9.6285 7.2975 5.2560 4.8690 (2, (2, (2, (2, 2) 2) 2) 2) 9.6778 7.3496 5.3082 4.9308 (2, (2, (2, (2, 2) 2) 2) 2) 0.2 10 3.0985 (3, 2.0070 (3, 1.1942 (3, 1.0243 (4, 2) 2) 2) 1) 3.1800 2.1120 1.3109 1.1483 (4, (4, (4, (4, 1) 1) 1) 1) 3.2097 2.1419 1.3497 1.1857 (4, (4, (4, (4, 1) 1) 1) 1) 6.5923 5.2586 3.3071 2.7666 (2, (2, (1, (1, 2) 2) 1) 1) 6.7395 5.4255 3.6690 3.1575 (2, (2, (1, (1, 2) 2) 1) 1) 6.7942 5.4763 3.7804 3.2803 (2, (2, (1, (1, 2) 2) 1) 1) 0.2 10 1.5636 1.0027 0.6067 0.5266 (3, (3, (3, (3, 1) 1) 1) 1) 1.7100 1.1723 0.7968 0.7176 (3, (3, (3, (3, 1) 1) 1) 1) 1.7615 1.2218 0.8488 0.7672 (3, (3, (3, (3, 1) 1) 1) 1) 2.9007 2.1341 1.4396 1.3004 (1, (1, (1, (1, 1) 1) 1) 1) 3.2760 2.5485 1.9020 1.7865 (1, (1, (1, (1, 1) 1) 1) 1) 3.3957 2.6758 2.0288 1.9180 (1, (1, (1, (1, 1) 1) 1) 1) 0.2 10 0.3204 0.1948 0.1285 0.1171 (1, (1, (1, (1, 1) 1) 1) 1) 0.7958 0.6194 0.5138 0.4980 (2, (2, (2, (2, 1) 1) 1) 1) 0.8613 0.6906 0.5905 0.5773 (2, (2, (2, (2, 1) 1) 1) 1) 1.3503 1.1552 1.0309 1.0236 (1, (1, (1, (1, 1) 1) 1) 1) 1.8405 1.6575 1.5315 1.5255 (1, (1, (1, (1, 1) 1) 1) 1) 1.9772 1.7740 1.6686 1.6607 (1, (1, (1, (1, 1) 1) 1) 1) 10 (plates) References ξ Reinforced Panel 0.8 0.6 0.4 0.2 R=3m, k=1, m=2, n=2 ξ0 =1e-5/h ξ =2e-5/h ξ =3e-5/h 0 0.65 0.75 τ 0.85 0.95 1.05 Fig Influence of initial imperfection on critical dynamic buckling load of reinforced panel Conclusions A formulation of the governing equations of eccentrically reinforced functionally graded cylindrical panels based upon the classical shell theory and the smeared stiffeners technique with von Karman–Donnell nonlinear terms has been presented By use of Galerkin method a nonlinear dynamic equation for analysis of dynamic and stability characteristics of ES-FGM cylindrical panels is obtained Fundamental frequencies of unreinforced and reinforced FGM panels are considered Some results were compared with the ones of other authors Nonlinear dynamic responses and critical dynamic loads of ESFGM cylindrical panels are investigated according to the criterion Budiansky–Roth They are significantly influenced by material parameters, stiffeners and initial geometrical imperfection Clearly, stiffeners enhance the stability and load-carrying capacity of FGM cylindrical panels Acknowledgements This paper was supported by the National Foundation for Science and Technology Development of Vietnam – NAFOSTED The authors are grateful for this financial support [1] Koizumi M The concept of FGM Ceram Trans Funct Grad Mater 1993;34:3–10 [2] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG Functionally graded materials: design, processing and applications London: Kluwer Academic Publishers; 1999 [3] Ng TY, Lam KY, Liew KM, Reddy JN Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading Int J Solids Struct 2001;38:1295–309 [4] Darabi M, Darvizeh 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analysis of plates Prentice-Hall; 1974 [30] Troitsky MS Stiffened plates Elsevier; 1976  ... stability of functionally graded stiffened cylindrical shells Appl Math Model 2009;54(2):179–307 [26] Dao Huy Bich, Vu Hoai Nam, Nguyen Thi Phuong Nonlinear postbuckling of eccentrically stiffened functionally. .. deflection of panel Hereafter, the couple of Eqs (16) and (17) or of Eqs (18) and (19) are used to investigate the nonlinear vibration and dynamic stability of panels They are nonlinear equations... 5029.2 5589.8 4.2 Vibration results To illustrate the proposed approach to eccentrically stiffened FGM cylindrical panels, the panels considered here are cylindrical panels and plates with in-plane